The Fundamental Theorem of Calculus

Let’s investigate the Fundamental Theorem of Calculus using programs written for the CASIO CFX-9850G.

Let’s look at two forms of this theorem.

First Form: Let f(x) be an integrable function defined on [a, b] and let F(x) also be defined on [a, b] such that F(x) is continuous on [a, b] and the derivative F(x) is f(x) for all x in (a,b). Then

ò ab f(x) dx = F(b) - F(a)

Second Form: Let f(x) be an integrable function defined on [a, b] and let F(x) also be defined on [a, b] by

F(x) = ò ab f(x) dx + F(a) for all x in [a, b]

where the real number F(a) is specified in advance. Then, F(x) is continuous on [a, b], and if f(x) is continuous at a point c in [a, b] then F(x) is differentiable at c also and F(c) = f(c)

We will use this second form and the CASIO CFX-9850G. to numerically verify the Fundamental Theorem of Calculus by doing the following:

(1) Choose an integrable (and continuous) function f(x) and convenient viewing window.

(2) Construct the numerical antiderivative F(x) = f(x) dx over an appropriate domain.

(3) Construct the numerical derivative of F(x) in (2) above.

(4) Graphically compare the numerical derivative of the numerical antiderivative of F(x) with the original function f(x) to see equality.

Obtain the following programs from the instructor by linking your calculators and transferring programs or key the programs below into the calculators memory:

Program FTC

ViewWindow -5, 5, 1, -5, 5, 1

Cls

Graph Y = Y1

StoPict 3

Cls

Seq(x, x, 1, 127, 1) --> List 1

List 1 -- List 2

"F(A) = " ? --> F

"A = " ? --> A

(Xmax - Xmin) / 127 --> D

1 --> B

Lbl 1

(Y1, A, Xmin+BD) --> C

Xmin+BD --> List1[B]

C --> List2 [B]

PlotOn Xmin+BD, C+F

Isz B

B < 126

Goto 1

StoPict 1

Program NDERIVE

Cls

List 1 --> List 3

For 2 --> I To 126

(List 2[I] - List 2[I-1]) / (List 1[I] - List 1[I-1]) --> List 3[I]

PlotOn List 1[I], List 3[I]

Next

StoPict 2

Stop

After receiving these two programs from us, follow these steps:

(2) Enter GRAPH mode and input the function Y1 = X2.

(3) Press MENU ALPHA log (B). You will see FTC and NDERIVE in the program list. Highlight FTC and press F1 (EXE) to execute this program.

(4) The program graphs Y1 in the viewing window defined by the program. After viewing, press EXE to continue program execution.

(5) You will be prompted for two input values.

At the prompt "F(A)=?", enter zero and press EXE.

At the prompt "A=?", enter zero and press EXE again.

(6) The screen is momentarily blank, but soon the numerical antiderivative of Y1 is plotted. (This plot is also saved in a picture memory that we will see later for comparison.)

(7) Press SHIFT EXIT (QUIT) MENU. Press 4 to see the x-values in List 1 and corresponding y-values in List 2 for the numerical derivative just plotted.

(8) After viewing these lists, press MENU ALPHA log (B). Highlight program NDERIVE and press F1 (EXE) to execute it. Soon, the numerical derivative is plotted from the data saved by program FTC into List 1 and List 2. This is the numerical derivative of the numerical antiderivative of Y1.

(9) Now press OPTN F1 (PICT) F2 (RCL) F1 (Pict1) to superimpose the numerical antiderivative of Y1.

(10) Press OPTN F1 (PICT) F2 (RCL) F3 (Pict3) to also superimpose the graph of Y1 on the screen. Look carefully.

Try these programs on the following functions. Be sure to first place into Y1 in GRAPH mode. In each case set, "F(A) = 0" and "A = 0".

(1) Y1 = x

(2) Y1 = x(x+1)(x-2)

Note: Trigonometric functions will not work well with program FTC as it is written. The View Window settings in the program must be appropriately set for these types of functions.

Dual Screen

A useful feature of the CASIO CFX-9850G is its ability to display split viewing screens so that different graphs can be viewed at the same time.

Use dual screen to view f(x) = x ( x + 2 )( x - 1 ) and g(x) = 1 - 2 x2

Turn on the CASIO CFX-9850G by pressing AC/ON.

Use the cursor arrows to highlight the GRAPH menu and press EXE or just press 5 when at the main menu screen.

Press SHIFT MENU (SETUP) and use the down cursor arrow to highlight the Dual Screen line. Press F1 (Grph). Press the EXIT key to return to the function list.

Press SHIFT F3 (V-Window) to set the parameters for the left screen. Use the following:

 Xmin = -3 Xmax = 3 Xscale = 0.5 Ymin = -3 Ymax = 3 Yscale = 1

Press F6 (RIGHT) to set the parameters for the right screen. Use the following:

 Xmin = -5 Xmax = 5 Xscale = 1 Ymin = -4 Ymax = 4 Yscale = 1

Press EXIT to return to the function list and enter into the Y1 slot X ( X + 2 )( X - 1 ) and into the Y2 slot 1 - 2 X2

Deselect Y1 by pressing F1 (SEL) and graph Y2 on the left screen by pressing F6 (DRAW).

Press OPTN F2 (SWAP). This interchanges the left screen which is the only active screen with the right screen which is inactive. Press EXIT to return to the function list and note the reverse video R appearing on the Y2 function line indicating that this is on the right screen.

Now select Y1 by highlighting if and pressing F1 (SEL). Press F6 (DRAW) to view the graph on the active left screen.

We can view each screen separately by pressing F6 (G<->T) for the active screen (the left screen) and F6 (G<->T) again for the inactive screen (the right screen).

Press EXIT or F6 (G<->T) to return to the function list. Press F6 (DRAW) to view both screens again and now press OPTN F1 (COPY). This superimposes Y2 on Y1 on the inactive screen.

Press EXIT to return to the function list and note that Y1 has a reverse video B on its function line indicating that it is plotted on both view windows.

Use the cursor arrows to highlight Y2 and press F2 (DEL) F1 (YES).

Use the cursor arrows to highlight Y1 and press F1 (SEL) to deselect Y1 removing it from both screens and press F1 (SEL) to select Y1 for graphing onto the active left screen.

Press F6 (DRAW) to see Y1 on the left screen.

The split screen feature is especially useful when using the other graphing features.

Press SHIFT F2 (ZOOM) F1 (BOX) and expand a box around the local maximum located at x = -2 -- this is accomplished by using the cursor arrows to move the "cross hairs" to a position marking the top left-hand corner of the desired box and pressing EXE then moving down to mark the bottom right-hand corner of the box by pressing EXE again. The desired box is magnified and place into the inactive right screen. Practice these steps for the local minimum at x = 1.

Press EXIT to return to the function list and F2 (DEL) F1 (YES) to delete Y1.

Turn off the CASIO CFX-9850G by pressing SHIFT AC/ON (OFF).

Graph to Table

Do a polar coordinate plot of r = 2 sin( q ) - cos( 2 q ) and store the values of the graph at 0° , 36° , 72° , 90° , 108° , 144° , 180° , 216° , 270° , 324° , and 360° into a table.

The Graph-to-Table feature of the CASIO CFX-9850G allows us to show both a graph and a table associated with the graph. We can move the "cross hairs" on a graph and store its values into a corresponding table.

Turn on the CASIO CFX-9850G by pressing AC/ON.

Use the cursor arrows to highlight the GRAPH menu and press EXE or just press 5 when at the main menu screen.

Press SHIFT MENU (SETUP) and use the down cursor arrow to highlight the Dual Screen line. Press F2 (GtoT). Continue to scroll down to the Angle line and press F1 (Deg). Press EXIT to return to the function list.

Press SHIFT F3 (V-Window) to set the parameters for the left screen. Use the following:

 Xmin = -2 Xmax = 2 Xscale = 0.5 Ymin = -1 Ymax = 4 Yscale = 0.25 Tmin = 0 Tmax = 360 Tpitch = 3.6

Press F3 (TYPE) F2 (r=) and enter 2 sin( q ) - cos( 2 q ) into slot r1. Press EXE to store the function.

Press F6 (DRAW). The graph appears on the left screen and an empty table appears on the right screen.

Press SHIFT F1 (Trace) and use the right cursor arrow to move the cross hairs to 0° . Press EXE to place the r and q values into the table on the right screen. Continue moving the cross hairs and storing the desired values.

The table appears as:

 r1 q 0 -1 36 0.8665 72 2.7111 90 3 108 2.7111 144 0.8665 180 -1 216 -1.484 270 -1 324 -1.484 360 -1

Do an xy-plot of y = x3 and y = x + 1 / x and create a table of their point of intersection and the local minimum of the second function.

Press F3 (TYPE) F1 (Y=) and enter X^3 into slot Y2. Press EXE to store the function.

Enter X + 1 ¸ X into slot Y3 and press EXE to store the function.

Press SHIFT F3 (V-Window) to set the parameters for the left screen. Use the following:

 Xmin = 0 Xmax = 4 Xscale = 0.5 Ymin = 0 Ymax = 4 Yscale = 0.5

Press F6 (DRAW). The graph appears on the left screen and an empty table appears on the right screen.

Use the Zoom Box feature to get a better view of the intersection. Press SHIFT F2 (ZOOM) F1 (BOX), move the cross hairs to the top left-hand corner of a desired box, and press EXE. Now move the cross hairs to the bottom right-hand corner of the desired box and press EXE. Press SHIFT F1 (Trace) and use the right cursor arrow to move the cross hairs to the point of intersection of Y2 and Y3. Use the up and down cursor arrows to view the point of intersection from the prospective of each function, for example from Y2 the intersection is (1.271592..., 2.056096...) and from Y3 the intersection is (1.271592..., 2.058007...)

Press EXE to place the X and Y values into the table on the right screen.

Press SHIFT F2 (ZOOM) F6 ( |> ) F1 (ORIG) to return to the original graph.

Press SHIFT F1 (TRACE) and use the up or down arrow to focus on Y3. Now use the right cursor arrow to move to the local minimum. Press EXE to store that point into the table on the right screen. The local minimum values are approximately (1.032258..., 2.001008...).

To save the X tabled data into List 1 press OPTN F2 (LMEM) F1 (List1).

Use the right cursor arrow to highlight the Y2 tabled data. To save it into List 2, press OPTN F2 (LMEM) F2 (List2).

Use the right cursor arrow to highlight the Y3 tabled data. To save it into List 3, press OPTN F2 (LMEM) F3 (List3).

Press EXIT MENU 4 and see that the table data has been transferred into these lists.

Turn off the CASIO CFX-9850G by pressing SHIFT AC/ON (OFF).

Dynamic Graphing

Animate the changes produced by the angle of the parametric graph given by

Xt = ( 10 cos a ) t and Yt = ( 10 sin a ) t - 4.9 t2

The Dynamic Graphing feature of the CASIO CFX-9850G allows us to display real-time representations of changes in a graph as coefficients are changed.

Turn on the CASIO CFX-9850G by pressing AC/ON.

Use the cursor arrows to highlight the DYNA menu and press EXE or just press 6 when at the main menu screen.

Press SHIFT MENU (SETUP) and use the down cursor arrow to highlight the Angle line. Press F1 (Deg). Press EXIT to return to the function list.

Deselect any functions in the function list by using the up or down cursor arrow to highlight the selected function and pressing F1 (SEL). We can also delete any functions that appear by again highlighting them and pressing F2 (DEL) F1 (YES).

Press SHIFT F3 (V-Window) to set the parameters. Use the following:

 Xmin = 0 Xmax = 15 Xscale = 2 Ymin = -1 Ymax = 4 Yscale = 0.25

Use the down cursor arrow to scroll down to enter the following:

Tmin = 0 Tmax = 2 Tpitch = 0.2

Press EXE to return to the function list. Press F3 (TYPE) F3 (Parm). Notice that the function list changes to accommodate parametric functions (Xt, Yt).

Enter into slot Xt1 the expression ( 10 cos A ) T. Note that the variable A is obtained by pressing ALPHA X,q ,T (A). Press EXE to store the function.

Enter into slot Yt1 the expression ( 10 sin A )T - 4.9 T2. Press EXE to store the function.

Press F4 (VAR), highlight A, enter the value 30, and press EXE. Press F2 (RANG) and set the parameters as follows:

 Start: 30 End: 75 pitch: 15

Press F3 (SPEED) and use the up or down cursor arrow to highlight the Normal line then press F1 (SEL). Press EXIT to return to the previous menu.

Press F6 (DYNA) to see the animation. This animation will continue through ten iterations or press AC/ON to stop the process at any point and return to the Dynamic Range screen. The speed of the presentation can be controlled from this screen. The menu selections are:

F1 ( |||> ) - Stop and go speed. With each EXE keystroke one graph is displayed.

F2 ( > ) - Slow speed presentation.

F3 ( |> ) - Normal speed presentation.

F4 ( >> ) - Fast speed presentation.

F5 (STO) - Stores graph settings and screen data.

F6 (DEL) - Delete Dynamic Graph screen data.

Animate the changes produced by the leading coefficient of the function

f(x) = b (x - 7)2 + 1.

Deselect any functions in the function list by using the up or down cursor arrow to highlight the selected function and pressing F1 (SEL). We can also delete any functions that appear by again highlighting them and pressing F2 (DEL) F1 (YES).

Keep the same viewing window settings as above.

Press F3 (TYPE) F1 (Y =) and enter into the Y2 slot the expression B ( X - 7 )2 + 1. Note that the variable B is obtained by pressing ALPHA log (B). Press EXE to store the function.

Press F4 (VAR), highlight B, enter the value 0.25, and press EXE. Press F2 (RANG) and set the parameters as follows:

Start: 0.25 End: 4.25 pitch: 1